Alexandra Shlapentokh

Defining integrality at prime sets of high density in number fields

1. Introduction. Interest in the questions of Diophantine definability and decid-ability goes back to a question that was posed by Hilbert: Given an arbitrary polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z? This question, otherwise known as Hilberts tenth problem, has been answered negatively in the work of M. (see [2] and [3]). Since the time when this result was obtained, similar questions have been raised for other fields and rings. Arguably the two most interesting and difficult problems in the area are the questions of Diophantine decidability of Q and the rings of algebraic integers of arbitrary number fields. One way to resolve the question of Diophantine decidability negatively over a ring of characteristic zero is to construct a Diophantine definition of Z over such a ring. This notion is defined below.

A ring version of Mazur's conjecture on topology of rational points

Remark 1.2. Let W be an algebraic set defined over a number field. Then W = V1∪· · ·∪Vk, k ∈ N, where Vi is a variety and W̄ = V̄1 ∪ · · · ∪ V̄k, with W̄, V̄1, . . . , V̄k denoting the topological closure of W, V1, . . . , Vk, respectively. Further, if nW , n1, . . . , nk are the numbers of connected components of W̄, V̄1, . . . , V̄k, respectively and ni < ∞ for all i = 1, . . . , k, then nW ≤ n1 + · · ·+nk. Thus, without changing the scope of the conjecture we can apply Conjecture 1.1 to algebraic sets instead of varieties.

Computable categoricity for algebraic fields with splitting algorithms

A computably presented algebraic field F has a splitting algorithm if it is decidable which polynomials in F [X] are irreducible there. We prove that such a field is computably categorical iff it is decidable which pairs of elements of F belong to the same orbit under automorphisms. We also show that this criterion is equivalent to the relative computable categoricity of F .

Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of Q

This paper provides the first examples of rings of algebraic numbers containing the rings of algebraic integers of the infinite algebraic extensions of Q, where Hilbert’s Tenth Problem is undecidable.Abstract This paper provides the first examples of rings of algebraic numbers containing the rings of algebraic integers of the infinite algebraic extensions of Q where Hilberts Tenth Problem is undecidable.

HILBERT'S TENTH PROBLEM FOR RINGS OF ALGEBRAIC FUNCTIONS IN ONE VARIABLE OVER FIELDS OF CONSTANTS OF POSITIVE CHARACTERISTIC

The author builds an undecidable model of integers with certain relations and operations in the rings of ^-integers of algebraic function fields in one variable over fields of constants of positive characteristic, in order to show that Huberts Tenth Problem has no solution there.

Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields

Abstract We prove that some infinite -adically discrete sets have Diophantine definitions in large subrings of number fields. First, if K  is a totally real number field or a totally complex degree-2 extension of a totally real number field, then for every prime of K  there exists a set of K-primes of density arbitrarily close to 1 such that there is an infinite -adically discrete set that is Diophantine over the ring of -integers in K. Second, if K  is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes of density 1 and an infinite Diophantine subset of that is v-adically discrete for every place v of K. Third, if K  is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes of density 1 such that there exists a Diophantine model of ℤ over . This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a nonarchimedean topology and questions concerning extensions of Hilbert’s Tenth Problem to subrings of number fields.

Categoricity properties for computable algebraic fields

We examine categoricity issues for computable algebraic fields. We give a structural criterion for relative computable categoricity of these fields, and use it to construct a field that is computably categorical, but not relatively computably categorical. Finally, we show that computable categoricity for this class of fields is

On Diophantine definability and decidability in some infinite totally real extensions of ℚ

\Pi^0_4

Diophantine Relations between Rings of

-complete.

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Let M be a number field, and W M a set of its non-Archimedean primes. Then let O M,WM = {x ∈ M|ord t x ≥ 0, ∀t ¬∈ W M }. Let {p 1 ,...,p r } be a finite set of prime numbers. Let F inf be the field generated by all the p j i -th roots of unity as j → ∞ and i = 1,...,r. Let K inf be the largest totally real subfield of F inf . Then for any e > 0, there exist a number field M ⊂ K inf , and a set W M of non-Archimedean primes of M such that W M has density greater than 1 - e, and Z has a Diophantine definition over the integral closure of O M ,W M in K inf .